Your friends are funnier than you are

Spoiler: How far your joke might reach depends on your friends going viral.

The spread of stories or beliefs are often compared to viruses and modeled as branching processes: one case becomes two, two becomes four, four becomes eight, and so on. But stories, like jokes, differ from branching processes in at least one key aspect: their 'infectiousness' can change as a result of the propagation itself. A joke might get funnier (or less funny) as it spreads, or a rumor might become more sensational with each retelling.

From a theoretical perspective, branching process mathematics has a surprisingly long history, beginning over 180 years ago with practical demographic concerns. In 1845, French mathematician Irénée-Jules Bienaymé studied aristocratic family extinction during Victorian times, developing the mathematical framework to answer whether noble bloodlines would survive. His work, largely forgotten, was later rediscovered through the famous collaboration between Francis Galton and Henry Watson in the 1870s, who posed the question: what proportion of family surnames would become extinct after many generations?

These early concerns about family names gave rise to a more mathematical question: if we start many independent cascades, what sizes will they reach? Some will die out immediately, others will grow modestly, and a few might explode into massive cascades. The distribution of these cascade sizes reveals deep patterns about how parts of the system scale together.

The mathematical language for describing these relationships is known as scaling laws. Throughout the 20th century, scaling concepts appeared across science: how organisms grow (allometric scaling), self-repeating patterns in nature (fractals), and computational complexity. But it was the explosion of digital social data in the 2000s that revealed how cascade size distributions in online systems follow these same scaling principles, opening entirely new questions about social contagion.

With branching processes, we focus specifically on cascade size distributions, which follow a remarkably simple mathematical relationship:

This expression states that the probability of a cascade of size s scales with exponent τ. A larger τ means a steeper decline—massive cascades become much rarer. This creates 'heavy-tailed' distributions, where, unlike human height or brain size which cluster around typical values, cascade sizes have no characteristic scale—there's always the possibility of something orders of magnitude larger. This is the mathematics of systems by which we're repeatedly surprised by events much larger than we expect.

Across vastly different systems—neural avalanches in the brain, forest fire sizes, earthquake aftershocks, and neutron chain reactions—the same specific value keeps appearing: τ = 3/2. Why this particular number? It emerges in systems poised at criticality, the precise mathematical point where branching processes balance between dying out and growing explosively. Systems with completely different mechanisms—neurons firing, trees burning, neutrons splitting—somehow converge on this same mathematical signature when they operate near this critical threshold.

The scaling exponent τ dramatically affects how cascades spread through networks:

But here's where reality deviates from theory. Empirical studies of social media cascades consistently show much steeper decay than the 3/2 prediction, with scaling exponents ranging from τ = 2 to 4. Twitter retweets, Facebook shares, and viral content all decay faster than critical branching processes predict, meaning massive viral events are rarer than the theory suggests. This systematic deviation points to something fundamental: real social cascades aren't simply branching processes with fixed rules. Instead, they hint at another mechanism lacking in classical branching processes: self-reinforcing cascade.

The model

In social media epidemics, one key difference is that the content of the cascade itself evolves as it spreads, leading to fundamentally new dynamics that break the assumptions of classical branching processes.

Unlike traditional epidemic models where a virus remains unchanged as it spreads, the Self-Reinforcing Model (SRC) model assumes that the 'content' can get stronger (reinforced) or weaker as it moves from person to person. Above, each concentric circle indicates a new generation $n$, with maximum cascade intensity $I_{max}$. Then, we have that

Unlike classical branching processes that predict explosive growth at criticality, self-reinforcing cascades create much steeper size distributions (higher τ values) because content quality varies. Weak content dies out quickly, while only exceptional content survives to create large cascades. This matches what we observe in social media, where most posts fade quickly while a few go viral.

This is why your friends are funnier than you are. Since most people are average funny, they tell jokes of moderate quality, which fails to create large cascade. A few people are exceptionally funny — their jokes (high intensity k) spread much further, so they reach many more listeners. When we ask ourselves how many people total are gonna hear the joke you might have told first, the odds are much higher that it came from one of those “exceptionally funny” people, simply because their jokes dominate the total audience.

Result

The log-log plot below demonstates that cascade size distributions follow power laws with different scaling exponents depending on the parameter values, with steeper power-law decays as you move further from criticality:

Below the critical point, the cascade size distributions of the SRC exhibits steep power-law tails with tunable scaling exponents. The key insight is that subcritical SRC are short but very broad compared to classic cascade models, meaning they produce power-law distributions even when not at criticality. This contrasts sharply with traditional cascade models that only exhibit power-law behavior at a precise critical point.

This result is exciting because it helps explain the ubiquity of power-law distributions in empirical social data, since real-world cascades don't need to be precisely tuned to a critical point to exhibit this behavior—they can show power-law scaling across a range of conditions.

Compare that to the classical branching process, where each person passes on the item (joke, belief, virus) to a random number of others, with some fixed reproduction distribution. At the critical point (where $R_0 = 1$) the size distribution of cascades is a power law. Away from criticality, the cascade size distribution has an exponential cutoff. That's why in ordinary branching models, heavy tails are fragile: they only appear exactly at criticality.

Conclusion

The bottom line is that, if the spread of jokes and stories is better described by SRC than classic branching process, than the person who told you a joke is likely one of the rare superspreader who made it funnier. And you on the other hand are probably an average person likely to make the joke worse. Statistically speaking, in these cascades, your friends are funnier than you are.